Let vec{a} = 5hat{i} - hat{j} - 3hat{k} and v | vedclass

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(D) The projection of vector $\vec{a}$ on vector $\vec{b}$ is given by the formula: $	ext{Proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\ve

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Projection of $\vec{a}$ on $\vec{b}$ is $\frac{-17}{\sqrt{35}}$ and the direction of the projection vector is opposite to the direction of $\vec{b}$

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(D) The projection of vector $\vec{a}$ on vector $\vec{b}$ is given by the formula: $	ext{Proj}_{\vec{b}} \vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$.Given $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$.First,calculate the dot product: $\vec{a} \cdot \vec{b} = (5)(1) + (-1)(3) + (-3)(5) = 5 - 3 - 15 = -13$.Next,calculate the magnitude of $\vec{b}$: $|\vec{b}| = \sqrt{1^2 + 3^2 + 5^2} = \sqrt{1 + 9 + 25} = \sqrt{35}$.The scalar projection is $\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|} = \frac{-13}{\sqrt{35}}$.Since the scalar projection is negative,the direction of the projection vector is opposite to the direction of $\vec{b}$.Note: The provided options contain a calculation error in the numerator ($17$ instead of $13$). Based on the sign of the dot product,the correct choice is $D$.

Let $\vec{a} = 5\hat{i} - \hat{j} - 3\hat{k}$ and $\vec{b} = \hat{i} + 3\hat{j} + 5\hat{k}$ be two vectors. Then which one of the following statements is $TRUE$?

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